Stylus on Nostr: Falling asleep the other night I was thinking about derivatives. I can think of two ...

Falling asleep the other night I was thinking about derivatives.

I can think of two functions that are their own derivatives: f(x) = 0 and f(x) = ex.

There's a pair of functions function such that each is its *second* derivative is itself: f(x) = e-x and g(x) = -e-x.

There's a group of functions such that its *fourth* derivative is itself, f(x) = sin(x), g(x) = cos(x), h(x) = -sin(x), k(x) = -cos(x)

Are there any functions with other numbers of derivative steps before they become equal to themselves again? 3, 5, 8, etc.

In the light of day, I think I see that we could define an "n-self-derivative function" by just setting the derivatives<code>f(0) = 1<br>f'(0) = -1/2<br>f''(0) = -1/2 <br>f'''(x) = f(x) etc<br></code>

and that defines a unique Maclurin series.

But .. are there functions of this form that would arise naturally?

#maths #math #mathstodon #recreationalMathematics